A Relationship between the Major Index for Tableaux and the Charge Statistic for Permutations

Kendra Killpatrick

Abstract

The widely studied $q$-polynomial $f^{\lambda}(q)$, which specializes when $q=1$ to $f^{\lambda}$, the number of standard Young tableaux of shape $\lambda$, has multiple combinatorial interpretations. It represents the dimension of the unipotent representation $S_q^{\lambda}$ of the finite general linear group $GL_n(q)$, it occurs as a special case of the Kostka-Foulkes polynomials, and it gives the generating function for the major index statistic on standard Young tableaux. Similarly, the $q$-polynomial $g^{\lambda}(q)$ has combinatorial interpretations as the $q$-multinomial coefficient, as the dimension of the permutation representation $M_q^{\lambda}$ of the general linear group $GL_n(q)$, and as the generating function for both the inversion statistic and the charge statistic on permutations in $W_{\lambda}$. It is a well known result that for $\lambda$ a partition of $n$, $dim(M_q^{\lambda}) = \Sigma_{\mu} K_{\mu \lambda} dim(S_q^{\mu})$, where the sum is over all partitions $\mu$ of $n$ and where the Kostka number $K_{\mu \lambda}$ gives the number of semistandard Young tableaux of shape $\mu$ and content $\lambda$. Thus $g^{\lambda}(q) - f^{\lambda}(q)$ is a $q$-polynomial with nonnegative coefficients. This paper gives a combinatorial proof of this result by defining an injection $f$ from the set of standard Young tableaux of shape $\lambda$, $SYT(\lambda)$, to $W_{\lambda}$ such that $maj(T) = ch(f(T))$ for $T \in SYT(\lambda)$.